Integrand size = 25, antiderivative size = 80 \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{f \sqrt {a+a \sin (e+f x)}} \]
-AppellF1(1/2,-n,1,3/2,1-sin(f*x+e),1/2-1/2*sin(f*x+e))*cos(f*x+e)*(d*sin( f*x+e))^n/f/(sin(f*x+e)^n)/(a+a*sin(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(80)=160\).
Time = 0.72 (sec) , antiderivative size = 242, normalized size of antiderivative = 3.02 \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \sin ^n(e+f x) (d \sin (e+f x))^n \left (-\sin ^2(e+f x)\right )^{-n} \sqrt {a (1+\sin (e+f x))} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (4 \operatorname {AppellF1}\left (-\frac {1}{2}-n,-\frac {1}{2},-n,\frac {1}{2}-n,\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (-\sin (e+f x))^n \sqrt {\frac {-1+\sin (e+f x)}{1+\sin (e+f x)}}-(1+2 n) \operatorname {AppellF1}\left (1,\frac {1}{2},-n,2,\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} \left (1-\frac {1}{1+\sin (e+f x)}\right )^n\right )}{4 a f (1+2 n) (-1+\sin (e+f x))} \]
(Cos[e + f*x]*Sin[e + f*x]^n*(d*Sin[e + f*x])^n*Sqrt[a*(1 + Sin[e + f*x])] *(4*AppellF1[-1/2 - n, -1/2, -n, 1/2 - n, 2/(1 + Sin[e + f*x]), (1 + Sin[e + f*x])^(-1)]*(-Sin[e + f*x])^n*Sqrt[(-1 + Sin[e + f*x])/(1 + Sin[e + f*x ])] - (1 + 2*n)*AppellF1[1, 1/2, -n, 2, (1 + Sin[e + f*x])/2, 1 + Sin[e + f*x]]*Sqrt[2 - 2*Sin[e + f*x]]*(1 - (1 + Sin[e + f*x])^(-1))^n))/(4*a*f*(1 + 2*n)*(-1 + Sin[e + f*x])*(-Sin[e + f*x]^2)^n*(1 - (1 + Sin[e + f*x])^(- 1))^n)
Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3266, 3042, 3265, 3042, 3264, 148, 333}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(d \sin (e+f x))^n}{\sqrt {a \sin (e+f x)+a}}dx\) |
\(\Big \downarrow \) 3266 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)+1} \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x)+1}}dx}{\sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)+1} \int \frac {(d \sin (e+f x))^n}{\sqrt {\sin (e+f x)+1}}dx}{\sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3265 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)+1} \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin ^n(e+f x)}{\sqrt {\sin (e+f x)+1}}dx}{\sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {\sin (e+f x)+1} \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin (e+f x)^n}{\sqrt {\sin (e+f x)+1}}dx}{\sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 3264 |
\(\displaystyle -\frac {\cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin ^n(e+f x)}{\sqrt {1-\sin (e+f x)} (\sin (e+f x)+1)}d(1-\sin (e+f x))}{f \sqrt {1-\sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 148 |
\(\displaystyle -\frac {2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \int \frac {\sin ^n(e+f x)}{\sin (e+f x)+1}d\sqrt {1-\sin (e+f x)}}{f \sqrt {1-\sin (e+f x)} \sqrt {a \sin (e+f x)+a}}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle -\frac {\cos (e+f x) \sin ^{-n}(e+f x) \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{f \sqrt {a \sin (e+f x)+a}}\) |
-((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1 - Sin[e + f*x])/2]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(f*Sin[e + f*x]^n*Sqrt[a + a*Sin[e + f*x]]))
3.2.30.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(d/b)^n*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a - x)^n*((2*a - x)^(m - 1 /2)/Sqrt[x]), x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n} , x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_), x_Symbol] :> Simp[(d/b)^IntPart[n]*((d*Sin[e + f*x])^FracPart[n ]/(b*Sin[e + f*x])^FracPart[n]) Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x ])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !I ntegerQ[m] && GtQ[a, 0] && !GtQ[d/b, 0]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Sin[e + f*x])^FracPart[m ]/(1 + (b/a)*Sin[e + f*x])^FracPart[m]) Int[(1 + (b/a)*Sin[e + f*x])^m*(d *Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 2, 0] && !IntegerQ[m] && !GtQ[a, 0]
\[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n}}{\sqrt {a +a \sin \left (f x +e \right )}}d x\]
\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
\[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int { \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx=\int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]